Turing Challenged

The inability to get past the 'fact' that there's no such thing as a real infinite series, thereby leading one to dismiss any argument using such a series (or similar such thing) as an interesting theoretical curiosity, but irrelevant to reality.

The problem goes away for the most part if one redefines an infinite series as something that could always use more space, no matter how big of a computer you got, where as a 'normal' series could potentially be calculated if one had a suitably big-but-finite number of atoms to work with.

--WilliamUnderwood

You mean infinite KolmogorovComplexity? impossible to describe the whole series no matter how you tried to write it down? Natural numbers are infinite, but only take a very, very small amount of space to fully describe: type N = 0 | s(N).

What is the connection with the title? With 'infinite series' I would have expected a title like TylorSeriesChallenged? or InfiniteSeriesDismissal?.

If the infinite calculations of a TuringMachine are meant (instead of mathematical infinite series), then just these TylorSeries? can be helpful too, because they are a finite representation of the infinite series.

--GunnarZarncke

It's a reference to the inability to determine for an arbitrary program if it's looping or simply not done yet, aka, the halting problem. A common objection is that the halting problem doesn't exist because for any possible real device, every state can potentially be enumerated, and therefore the halting problem is 'solvable' on that device.

The objection is sound in theory, but in practice MooresLaw renders the set of potential states to be countably infinite.